On Cellular Straight Line Segments.

Abstract
We discuss a scheme for digitizing curves that is consistent with a scheme for digitizing regions. It is shown that the cellular image of a region is determined by the cellular image of its boundary by the scheme. It is proved that the chord property is a necessary and sufficient condition for a cellular arc to be a cellular straight line segment. By showing that the chord property and the cellular convexity condition are equivalent, we prove that a cellular arc is a cellular straight line segment if and only if it is cellularly convex. This leads to an algorithm to determine whether or not a cellular complex is a cellular straight line segment in time linear in the number of rows of cells. Finally it is proven that a cellular complex is cellularly convex if and only if any pair of its cells is connected by a cellular straight line segment in the cellular complex. (Author)

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